What is the value of sin(2arctan(sqrt(2))) ?

The value of sin(2arctan(sqrt(2))) is (2sqrt(2))/3

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sin(2arctan(sqrt(2)))

sin (2 arctan (2))

\frac{2 2}{3}

Decimal Notation

0.94280 \dots

Solution steps

sin (2 arctan (2))

Rewrite using trig identities:

2 sin (arctan (2)) cos (arctan (2))

sin (2 arctan (2))

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (arctan (2)) cos (arctan (2))

= 2 sin (arctan (2)) cos (arctan (2))

Rewrite using trig identities:

sin (arctan (2)) = \frac{6}{3}

sin (arctan (2))

Rewrite using trig identities:

sin (arctan (2)) = \frac{2 1 + ( 2 ) ^{2}}{1 + ( 2 ) ^{2}}

Use the following identity:

sin (arctan (x)) = \frac{x 1 + x ^{2}}{1 + x ^{2}}

= \frac{2 1 + ( 2 ) ^{2}}{1 + ( 2 ) ^{2}}

= \frac{2 1 + ( 2 ) ^{2}}{1 + ( 2 ) ^{2}}

Simplify

= \frac{6}{3}

Rewrite using trig identities:

cos (arctan (2)) = \frac{3}{3}

cos (arctan (2))

Rewrite using trig identities:

cos (arctan (2)) = \frac{1 + ( 2 ) ^{2}}{1 + ( 2 ) ^{2}}

Use the following identity:

cos (arctan (x)) = \frac{1 + x ^{2}}{1 + x ^{2}}

= \frac{1 + ( 2 ) ^{2}}{1 + ( 2 ) ^{2}}

= \frac{1 + ( 2 ) ^{2}}{1 + ( 2 ) ^{2}}

Simplify

= \frac{3}{3}

= 2 \cdot \frac{6}{3} \cdot \frac{3}{3}

Simplify

2 \cdot \frac{6}{3} \cdot \frac{3}{3} : \frac{2 2}{3}

2 \cdot \frac{6}{3} \cdot \frac{3}{3}

Multiply fractions:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{6 3 \cdot 2}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{2 6 3}{9}

Factor

9 : 3^{2}

Factor

9 = 3^{2}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

= 23

= \frac{2 2 3 3}{3 ^{2}}

233 \cdot 2 = 2 \cdot 3 \cdot 2

233 \cdot 2

Apply radical rule:

a a = a

33 = 3

= 3 \cdot 22

= \frac{3 \cdot 2 2}{3 ^{2}}

Cancel the common factor:

3

= \frac{2 2}{3}

= \frac{2 2}{3}

\frac{tan ( 7 2 ^{\circ} ) - tan ( 1 2 ^{\circ} )}{1 + tan ( 7 2 ^{\circ} ) tan ( 1 2 ^{\circ} )}

arcsin (0.88)

0 csc (0)

10 sin (15 0^{\circ})

cos (6 5^{\circ}) + cos (5 5^{\circ})

What is the value of sin(2arctan(sqrt(2))) ?
The value of sin(2arctan(sqrt(2))) is (2sqrt(2))/3